The abelian gauge-Yukawa $\beta$-functions at large $N_f$

We study the impact of the Yukawa interaction in the large-$N_f$ limit to the abelian gauge theory. We compute the coupled $\beta$-functions for the system in a closed form at $\mathcal{O}(1/N_f)$.


Introduction
A comprehensive understanding of the UV behaviour of gauge-Yukawa theories has become of key importance with the growing interest in the asymptotic-safety paradigm [1][2][3][4]. Prime candidates for these considerations are gauge-Yukawa models with a large number of fermion flavours, N f . Computing the leading large-N f contribution to the β-functions was pioneered by evaluating the O(1/N f ) gauge β-functions [5][6][7] for N f fermions charged under the gauge group; see also Refs [8,9].
We recently computed the O(1/N f ) β-function for Yukawa-theory [10] inspired by the earlier works [11,12]. The Yukawa-theory is closely related to the Gross-Neveu model, which has been extensively studied in the past using a different approach; see e.g. Refs [13][14][15][16]. For Gross-Neveu-Yukawa model the behaviour near the fixed point in terms of critical exponents is known up to O(1/N 2 f ) [17,18]. However, the strength of our analysis is that we readily achieved a closed form expression of the β-function, and as shown in the present work, the procedure is straighforwardly generalisable to include gauge interactions.
In this paper, we compute the leading 1/N f contributions to the β-functions of the gauge-Yukawa system in a closed form. This result is new and sheds light to the impact of the Yukawa interaction to the gauge theory in the large-N f limit.
The gauge contribution to the Yukawa β-function was computed in the abelian case in Ref. [11] and later generalised to non-abelian and semi-simple gauge groups in Ref. [12] assuming that only one flavour of fermions couples to the scalar via Yukawa interaction. We relax this assumption and show that it is possible to get a closed form expressions also in the general case. The current result provides a groundwork for several interesting extensions including e.g. non-abelian gauge groups and chiral fermions.
The paper is organized as follows: In Sec. 2 we introduce the framework and notations and in Sec. 3 compute the new contributions to the renormalization constants and βfunctions. In Sec. 4 we collect the results and comment on the structure of the coupled system, and in Sec. 5 we conclude.

The framework
We consider the massless U(1) gauge theory with N f fermion flavours (QED) with a gaugesinglet real scalar field coupling to the fermionic multiplet, ψ, via Yukawa interaction: We define the rescaled gauge and Yukawa couplings, which are kept constant in the limit N f → ∞. The purpose of this paper is to derive the coupled system of β-functions for E and K at the 1/N f level: where G 1 and H 1 are defined by and Z 3 , Z S , Z F , and Z V are the renormalization constants for the photon, the scalar, and the fermion wave function, and the 1PI vertex, respectively. The photon wave function renormalization constant, Z 3 , is given by where Π 0 is the self-energy divided by the external momentum squared, p 2 , and we denote the poles of X in by divX. The self-energy can be written as E is the one-loop contribution, and Π K contain the n-loop part consisting of n − 2 fermion bubbles in the gauge and Yukawa chains summing over the topologies given in Fig. 1.
The scalar wave function renormalization constant, Z S , is determined via with the scalar self-energy given by K is the one-loop result, and S For the fermion self-energy and vertex renormalization constants, the lowest non-trivial contributions are already O(1/N f ), and we have where Σ contain n − 1 fermion bubbles and are shown diagrammatically in Fig. 3b.
The term corresponding to pure QED, Π (n) E , was computed in Ref. [6], and the pure- K , in Ref. [10]. Their contribution to the βfunctions, Eqs (2.3) and (2.4), is  where The impact of the mixed contributions, namely Π (n) E , is evaluated in the next section.

Mixed contributions
In this section we derive the mixed contributions to the renormalization constants for the photon self-energy, the fermion self-energy, the Yukawa vertex, and the scalar self-energy, and eventually compute the coupled β-functions.

The Yukawa contribution to the QED β-function
The Yukawa contribution to the photon self-energy (depicted in the second row of Fig. 1), is obtained by substituting Eq. (2.8) in Eq. (2.7). We get Notice that the diagrams involving a horizontal bubble chain differ from the corresponding ones for the scalar self-energy in Fig. 2 just by an overall factor (2 − d) coming from the algebra of the γ-matrices. Altogether, we find where π K (p 2 , , n) can be expanded as where we used and restricted ourselves to the 1/ pole. The function π (0) K is independent of p 2 , as it should, and reads π (0) The contribution of Z 3 (K) to β E , Eq. (2.3), is found to be where we have defined We show the function π K (t) in Fig. 4. Since π K (t) has a first order pole at t = 3, the first singularity of β E (K = 0) occurs at K = 3 and is a logarithmic one. The next singularity of π K (t) is found at t = 5 (first order) and would result in a logarithmic singularity of β E (K = 0) at K = 5.

The QED contribution to the Yukawa β-function
The QED contribution to the fermion self-energy and to the Yukawa vertex is closely related to the pure-Yukawa case. This is because the gauge chain is equivalent to the Yukawa chain besides an overall factor. In fact, Σ (n) and (3.14) Using the one-loop result Z E = 1 − 2 3 E −1 + O(1/N f ), and applying the same summation procedure as in Ref. [10] for the fermion self-energy and the vertex, Eqs (2.11) and (2.13) where we kept only the 1/ pole. The functions σ E are independent of p 2 , and are given by The QED contribution to the scalar self-energy is shown in the second row of Fig. 2. The diagrams involving a horizontal gauge chain are related to the ones in the pure-Yukawa case analogoursly to Eq. (3.9). Altogether, we find The QED contribution in Eq. (2.9) is singled out as follows: To evaluate the right-hand side of Eq. (3.21), we closely follow the procedure in Ref. [10] for the scalar self-energy: where we defined ξ E (p 2 , , n) = (n + 1) 2 S and using we can further simplify the expression to where we kept the 1/ pole only. The function ξ E (p 2 , t, 0) = lim n→0 ξ E (p 2 , t, n) has to be independent of p 2 for the consistency of the computation. This is indeed the case: we checked that and therefore ξ E (p 2 , t, 0) = −2v (3.28) Finally, we find: With Eqs (3.15), (3.16) and (3.29) at hand, we can compute the QED contribution to the Yukawa β-function: where we have definedξ The functions ξ E (t) andξ E (t) are explicitly given by We plot the functionsξ E (t) and ξ E (t) in Fig. 5. The first singularity of β K (E = 0) is at E = 15/2 and consists of a first-order pole coming from ξ E (t) plus a logarithmic singularity arising from the integration ofξ E (t), both at t = 5.

The coupled system
Here we summarize and discuss our results for the coupled system. Combining Eqs (2.15) and (2.16) with the new results in Eqs (3.7) and (3.30), we obtain Near the Gaussian fixed point, these can be expanded as  We have checked that the expansions agree with the known four-loop results [19][20][21][22][23] in the leading order in N f . Furthermore, the − 2E 3K ξ E corresponds to the result of Refs [11,12], and we have checked that our result agrees with those.
The first singularity of the pure-QED β-function is located at E = 15/2, whereas for the pure-Yukawa case it occurs at K = 5. These known singularities are now accompanied by the ones from the mixed contributions, Eqs (3.7) and (3.30). As we noticed in Section 3, β E (K = 0) has the first singularity at K = 3, while β K (E = 0) at E = 15/2. The former, similarly to the pure gauge and Yukawa cases, is a logarithmic singularity, whereas the latter is a pole of first order.
The O(1/N f ) coupled system has only the three already known fixed points: the Gaussian fixed point, and the pure-QED (near E = 15/2) and pure-Yukawa (near K = 3) fixed points.
We show the flow diagram for N f = 30 outside the vicinity of the singularities in Fig. 6. Near K = 3, the logarithmic singularity in β E arising from π K (t) dominates making the gauge coupling to increase and approach the value E = 15/2. Near E = 15/2, however, β K has a pole arising from ξ E (t) eventually dominating the flow, and driving the Yukawa coupling to zero near E = 15/2. The flow may be extended setting K ≡ 0 and switching to pure-QED, so that the gauge coupling reaches the fixed point as E → 15/2 in the UV.

Conclusions
We have computed the leading 1/N f mixed contributions for the β-functions for abelian gauge-Yukawa theory with N f fermion flavours coupling to a gauge-singlet real scalar. Together with the known results for the pure-QED and pure-Yukawa cases, this allows the study of the abelian gauge-Yukawa system.
The flow in the interacting theory leads to the vanishing Yukawa coupling near the gauge coupling value E = 15/2 due to the peculiar interplay of the singularities. However, the gauge β-function is still positive around (K, E) = (0, 15/2), and E keeps growing before eventually reaching the fixed point due to the known a logarithmic singluarity near E = 15/2.
Our work extends the previous results towards a more complete picture of gauge-Yukawa theories in the large-N f limit.